The Feynman-Kac formula establishes a link between partial differential equations (PDEs) and stochastic processes. It offers yet another method of solving certain PDEs: by simulating random paths of a stochastic process.
Suppose we are given the PDE
subject to the terminal condition
u(x,T) = ψ(x)
where μ, σ2, ψ are known functions and u is the unknown. Then FK tells us that the solution can be written as an expectation:
u = E[ψ(XT) | X = X0]
where X is an Itô process driven by the equation
This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods
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